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Polynomial Averages Converge to the Product of Integrals

Nikos Frantzikinakis·Bryna Kra
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Abstract

We answer a question posed by Vitaly Bergelson, showing that in a totally ergodic system, the average of a product of functions evaluated along polynomial times, with polynomials of pairwise differing degrees, converges in $L^{2}$ to the product of the integrals. Such averages are characterized by nilsystems and so we reduce the problem to one of uniform distribution of polynomial sequences on nilmanifolds.

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Nikos Frantzikinakis, Bryna Kra. 2004-03-26. Polynomial Averages Converge to the Product of Integrals. https://arxiv.org/abs/math/0403454

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